8/6/2019 CDS_JPM1

1/1

New York

October 24, 2001

JPMorgan Securities Inc.

Credit Derivatives

1Our pricing analytic is available on Orbit ( www.morgancredit.com) or on Bloomberg ([ticker] [coupon] [maturity]CDSW ).

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Purpose: There have been considerable client inquiries on how

default probabilities are calculated from credit default swap spreads

using our pricing analytic1. This is a quantitative process that is not

easy to explain intuitively. As such, rather than explain the

calculation of default probabilities from credit default swap spreads,

this paper focuses on the reverse approximating par credit default

swap spreads from default probabilities.

Definition: A credit default swap is an agreement in which one party

buys protection for losses occurring due to a credit event of a

reference entity up to the maturity date of the swap. The buyer of

protection on a bond, a loan, or a class of bonds or loans, will

typically buy protection on the notional of the asset and, upon the

occurrence of a credit event, would deliver an obligation of the

reference credit in exchange for the protection payout. The

protection buyer pays a periodic fee for this protection up to the

maturity date, unless a credit event triggers the contingent payment.

If such trigger happens, the buyer of protection only needs to pay

the accrued fee up to the day of the credit event (standard credit

default swap).

Determining the Par Spread: A credit default swap has two

valuation legs: fee and contingent. For a par spread, the net present

value of both legs must equal to zero.

The valuation of the fee leg is approximated by:

== NN AnnuitySPmtsFeeDefaultNoofPV

=

N

i

iiiN PNDDFS1

where, SN is the Par Spread for maturity N

DFi is the Riskless Discount Factor from To to TiPNDi is the No Default Probability from To to Tii is the Accrual Period from Ti-1 to Ti

If accrual fee is paid upon default, then the valuation of the fee leg is

approximated by:

=+ AccrualsDefaultofPVPmtsFeeDefaultNoofPV

=+ NNNN AccrualDefaultSAnnuityS

( ) = =

+

N

i

N

i

iiiiNiiiN PNDPNDDFSPNDDFS

1 1

12

where, (PNDi-1 PNDi) is the Probability of a Credit Event

occurring during period Ti-1 to Ti

2

i

is the Average Accrual from Ti-1 to Ti

The valuation of the contingent leg is approximated by:

==N

ContingentContingentofPV

( )=

N

i

iii PNDPNDDFR1

1)1(

where, R is the Recovery Rate of the reference obligation

Therefore, for a par credit default swap,

LegContingentofValuationLegFeeofValuation =

or

( )

+ = =

N

i

N

i

iiiNiiiNPNDPNDDFSPNDDFS

1 1

1

( )

=

N

i

iii PNDPNDDFR1

1)1(

or

( )

( )

=

=

+

=N

i

iiiiii

N

i

iii

N

PNDPNDDFPNDDF

PNDPNDDFR

S

1

1

1

1

2

)1(

Example:

Recovery 30%

Period (i) Yldi DFi PNDi AnnuityN

Default

AccrualN ContingentN

App

(A

0.25 2.35% 99.41% 96.43% 0.240 0.004 0.025 1

0.5 2.33% 98.84% 93.05% 0.470 0.008 0.048

0.75 2.39% 98.25% 89.76% 0.690 0.012 0.071

1 2.52% 97.63% 86.56% 0.901 0.016 0.093

1.25 2.70% 96.98% 83.91% 1.104 0.019 0.111

1.5 2.87% 96.28% 81.51% 1.300 0.022 0.127

1.75 3.05% 95.55% 79.41% 1.490 0.025 0.141

2 3.22% 94.78% 77.30% 1.673 0.028 0.155

2.25 3.37% 93.99% 75.79% 1.851 0.030 0.165

2.5 3.52% 93.17% 74.32% 2.024 0.032 0.175

2.75 3.67% 92.31% 72.87% 2.192 0.034 0.184

3 3.82% 91.44% 71.45% 2.355 0.036 0.193

3.25 3.92% 90.55% 70.66% 2.515 0.037 0.198

3.5 4.02% 89.64% 69.90% 2.672 0.038 0.203

3.75 4.12% 88.72% 69.14% 2.825 0.039 0.208

4 4.22% 87.79% 68.37% 2.975 0.040 0.213

4.25 4.30% 86.85% 68.22% 3.123 0.040 0.214 4.5 4.37% 85.91% 68.06% 3.269 0.040 0.215

4.75 4.45% 84.96% 67.91% 3.413 0.040 0.216

5 4.52% 84.00% 67.76% 3.555 0.040 0.217

Par Credit Default Swap Spread Approximation from Default Probabilities